Or is the trauma of losing a child mainly social, and society was adapted to cope with it on an individual level in a way that people weren't mentally damaged by it in a serious and irreversible way (that would greatly impact their behaviour and mental health, like today)?
I have an earlier answer that--surprise!--looks at Christian and Jewish experiences in the Middle Ages:
How did parents dead with high child and infant mortality in the Middle Ages?
Parents--mothers and fathers alike--took the deaths of their children quite hard. This might seem like a "duh" statement, but for a long time, there was a prominent strand of medievalist scholarship that argued the opposite--high mortality rates meant parents refused to invest in their children emotionally until somewhere in toddlerhood. More recent work has demonstrated this to be as a blanket statement utterly untrue. Elisheva Baumgarten turned up legal/religious advice cases from the Rhineland (Ashkenaz) Jewish community that implore Jewish fathers to stop crying so much and acting like women over the deaths of their children.
As for coping? There's one completely heartbreaking ritual/miracle legend that arose specifically in the Latin Christian community for stillborn babies or babies who died during/right after birth. In medieval Christian theology, humans needed baptism to cleanse them of original sin; otherwise they were bound for hell if they died. Yes, completely innocent babies (since they were not actually innocent). Various emergency measures developed to deal with this, such as authorizing midwives to perform the sacrament of baptism in the most dire cases. But sometimes all really was lost. So the "miraculous resurrection" developed--the midwife, or may even one of the parents, would claim the stillborn child suddenly came to life! JUST long enough for it to be taken to a nearby shrine and baptized, at which point God took its soul securely to heaven.
Both Jewish and Christian religious teaching also used the language and stories of martyrdom to try to teach parents to cope with potential grief over lost children. In Judaism, attention focused on the women martyrs killed by the First Crusaders in their pogroms across Germany. These women were renowed not just for "sanctifying the name of the Lord" themselves, but for putting their own sons and daughters, teenagers and infants, to the sword first so the Christian knights would not kidnap them, have them baptized, and raise them as Christian. Parents were encouraged to see a greater religious purpose in the loss of children.
Christians relied on a standard trope in saints' and martyrs' lives: the need to leave (living) children behind in order to worship God to the best of their ability. This might mean Perpetua breastfeeding her son one last time before joyfully going to her martyrdom in the arena; it might mean the litany of women saints who left their children with their own parents and wandered into the desert to become monks. Medieval religious teaching did not encourage imitation of these behaviors. It encouraged emulation of the ideal of separation from earthly attachment to focus on God.
As a follow up, would the impact be lower in close-knit tribes of hunter gatherers with better social support than would be found in agriculturally-based civilizations with greater social stratification and more physical/socio-cultural barriers to group intimacy?
And what role do certain beliefs regarding babies/souls/etc. play in various groups/cultures?
As a follow up, is it really inherently damaging? Are there people who can just move on without really being affected?